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I have been tutoring linear algebra, and my student does not seem to be able

to understand a solution I proposed ( of course, I may be wrong, and/or explaining

poorly). I'm hoping someone can suggest a better explanation and/or a different solution

to this problem:

We have two vector spaces V,V' , over the same F, and we have a subspace S of V.

The goal is to construct a map T , whose kernel is precisely S. The dimensions of

S,V,W respectively work well re Rank-Nullity, i.e., DimV-DimS=DimV'. My goal is to

declare T to be zero in B_S , and then set up a bijection between the remaining basis

vectors in B_V , and the basis vectors in B_V'.

So, I propossed that:

i)We choose a basis B_S :={e_1,...,e_s} for S, extend to a basis B_V:= {e_1,e_2,...,e_b}

for V. Let B_W:={e_1',e_2',...,e_w'} ; s:=|B_S|, and so-on.

ii)Declare/define T(B_S)==0 , i.e., for each basis vector e_s in B_S, we define

T(e_s)=0

iii) Now, we set up a bijection between the basis vectors in B_V\B_S , and those in

B_W. This bijection, gives rise to an isomorphism (extending by linearity) between

Span(B_V-B_S) , and V' , so we have:

1)T(e_s)==0 , for e_s in B_S

2)T(e_s+i):=e_i'

3)T(ce_s+i+de_s+j):=cT(e_s+i)+dT(e_s+j)

Now, I'm trying to extend this to the infinite-dimensional case, but my student has

only a beginners' knowledge of set theory.

Any Suggestions?

Thanks in Advance.

P.S: She also asked me a sort-of-strange question: is there such a thing as "non-linear

algebra"? I had no idea how to answer that. anyone Know?